>> >> Pam's steps were...>> >> 1. They would draw a horizontal rectangular bar,>> divide it into 5 equal parts.>> 2. lightly shade 3>> 3. divide each of the remaining 2 parts in half>> 4. (as well as the original 3 so as to continue to>> have equal portions)>> 5. By shading 1/4 of the "afternoon tarts" (1/10 of>> the whole bar)>> 6. and shading an equal portion of the "morning>> tarts">> 7. what is left of the morning tarts is equal to 200>> 8. Since there are 5 parts, each part is equal to 40>> >> Where did steps 3 and 4 come from? and step 6? Think>> like a child that doesn't know how to solve this>> problem. You can't get those steps without already>> having rationalized the solution mathematically.>> > > OK, let's try this (putting my teacher hat on, although if I dwell too long in Bob's mind, I fear I will get claustrophobic).> > Bob, age 10, makes a pizza with his mom. He has invited 4 friends, plus himself, so he cuts the pizza into 5 equal pieces. 2 friends arrive at noon, and they and Bob eat their slice of pizza. So at 12:15, 3/5 of the pie is gone and 2/5 remain. The other two friends arrive at 1:00, but they have each unexpectedly brought a friend. What is Bob to do now that he needs to feed 4 more friends? He realizes he can cut each of the remaining 2 pieces in half. Did he need to know this at 11:45 when he originally cut the pizza? Did he need to reverse engineer? No, and he doesn't even need to know at this point that each of the 1:00 friends are eating 1/10 of the whole pizza, only that they are each getting 1/4 of the remainder.> > One of the later boys is really hungry, so before eating his too-small slice, he starts to wonder how much more he would have had if he could have eaten all of what was eaten at noon. From all the comparison problems he solved in 3rd grade, he knows he must subtract his portion from the noontime portion to find the difference. Luckily, the knife left marks in the pizza pan where it was sliced, so he laid his slice next to one of the marks and he can see the difference, but how to name it? The pieces are different sizes. Then he remembers that his slice is half of each original slice, so he uses the knife to mark the pan accordingly, still not knowing a "number answer" to his question (so still no reverse engineering required). Ah, now he sees he would have had 5 more pieces the size of his own.> > Here comes the reverse engineering and it comes only from the original problem poser, not the solver. Mom (who loves counting) knows she and Bob placed 400 shreds of cheese on the whole pizza, so she tells the boy that he got 200 shreds of cheese less than the three noon boys together, and asks him how much cheese was on the whole pizza. He knows that 5 pieces the size of his own therefore had 200 shreds of cheese, and since mom said she carefully placed the same number on each piece, he simply divides, then multiplies by 10.> > Does that help?> > Pam

I thought this was going to end with "Who was driving the bus?":)

> Bob, age 10, makes a pizza with his mom. He has invited 4 friends, plus himself, so he cuts the pizza into 5 equal pieces. 2 friends arrive at noon, and they and Bob eat their slice of pizza. So at 12:15, 3/5 of the pie is gone and 2/5 remain. The other two friends arrive at 1:00, but they have each unexpectedly brought a friend. What is Bob to do now that he needs to feed 4 more friends? He realizes he can cut each of the remaining 2 pieces in half. Did he need to know this at 11:45 when he originally cut the pizza? Did he need to reverse engineer? No, and he doesn't even need to know at this point that each of the 1:00 friends are eating 1/10 of the whole pizza, only that they are each getting 1/4 of the remainder.

This was played out in real time. The only math was at the end, divide two pieces into 4 pieces. This is not what I meant by "reverse engineering".

> One of the later boys is really hungry, so before eating his too-small slice, he starts to wonder how much more he would have had if he could have eaten all of what was eaten at noon. From all the comparison problems he solved in 3rd grade, he knows he must subtract his portion from the noontime portion to find the difference. Luckily, the knife left marks in the pizza pan where it was sliced, so he laid his slice next to one of the marks and he can see the difference, but how to name it? The pieces are different sizes. Then he remembers that his slice is half of each original slice, so he uses the knife to mark the pan accordingly, still not knowing a "number answer" to his question (so still no reverse engineering required). Ah, now he sees he would have had 5 more pieces the size of his own.

Is this an imaginary kid? 2/5 of the pizza remains, 3/5 was eaten, he has 1/10, 3/5 - 1/10 = 5/10. You then reverse engineered an imaginary story from that. If kids were that crazy deductive, wouldn't they have gotten 3/5 - 1/10 long ago?

> Here comes the reverse engineering and it comes only from the original problem poser, not the solver. Mom (who loves counting) knows she and Bob placed 400 shreds of cheese on the whole pizza, so she tells the boy that he got 200 shreds of cheese less than the three noon boys together, and asks him how much cheese was on the whole pizza. He knows that 5 pieces the size of his own therefore had 200 shreds of cheese, and since mom said she carefully placed the same number on each piece, he simply divides, then multiplies by 10.

Yes, this is also reverse engineering (mom creating the problem). That is how (most) problems are created, but not solved. They are solved in the manner you presented in this third paragraph. Systematically and heuristically.